in any triangle ABC prove that 1/sin^nA+1/sin^nB+1/sin^nC>=3*2^n

The question is incorrect. Try it for n=1 and A= B=C= 60

Then the equation would be 1/sinⁿA  + 1/sinⁿB + 1/sinⁿC = 3 * (2/root(3))^1 =  root(3) 2 which is no doubt smaller than 3* 2^n

This equation is valid only for n < 0

Proof:

Using AM- GM inequalitites 

(1/sinⁿA  + 1/sinⁿB + 1/sinⁿC)/3 >= (1/sinⁿA  * 1/sinⁿB * 1/sinⁿC )^1/3

With the equality existing only when 1/sinⁿA  = 1/sinⁿB = 1/sinⁿC

This can only be possible when sin A = sin B = sin C = 60

Then the minimum value of 1/sinⁿA  + 1/sinⁿB + 1/sinⁿC is ((2/root(3))³ⁿ )^1/3

= ((2/root(3))ⁿ

= ((2ⁿ/root(3)ⁿ)*3

= (2ⁿ/3^n-2/2)

now if (n-2)/2 is less than -1 or we can say n < 0 then the question is no doubt correct.

The correct question would be 1/sinⁿA  + 1/sinⁿB + 1/sinⁿC>= (2ⁿ/3^n-2/2) or

1/sinⁿA  + 1/sinⁿB + 1/sinⁿC>= (3*2ⁿ) where n<0

no....the question is for n>=1 and n belogs to N

Try it yourself with A=B=C=60 and n=1,2,3,4,...

You would surely find that the question is errorenous. Just try ones and u wil find that the question is true only for n<0.  

but you have computed the answer taking only the equality into consideration

and it is not only for A=B=C=60.............should work for any triangle

OK... Then try it for a combination of 90,60,30. Is that Ok...

The result is 3 + 2/root(3) which is surely smaller than 6(3*2^1)

Now iv proved it fr inequality. 

proof for inequality.....not a verification

Shrihari,

Iv already given u a proof using Am-Gm inequalitites in the first reply but if thr is something which u cant understand u can ask me. Certainly i m able to clear ur confusions but fr tht u hv to clr me what is ur opinion fr ur question . wht do you think the mistake is. Just try to look at my first reply .

Fr understading my reply, u should first read about Am-Gm inequalities. I think u will certianly get tht my proof not only cover the case with equal angle but also with unequal angles. Till now im inable to understand your prob...